1/26 On Pareto conjugate priors and their application to large claims reinsurance premium calculation. José L. Vilar-Zanón Departamento de Economía Financiera y Actuarial. Universidad Complutense de Madrid Cristina Lozano-Colomer Departamento de Métodos Cuantitativos Universidad Pontificia de Comillas (ICADE) Abstract: This paper addresses the Bayesian estimation of the shape parameter of Pareto distributions, and its application to premium calculation of large claims XL reinsurance contracts. It studies the use of the Generalized Inverse Gaussian (GIG) as a Pareto prior conjugate, a family that contains as a particular case the Gamma distribution. An Exact Credibility formula is deduced allowing the calculation of individual reinsurance premiums. These are premiums suited to the excesses history of a sole portfolio. A family of predictive distributions for the excesses is derived. We apply our Exact Credibility model to a sample of excesses arisen in ten Spanish portfolios of liability motor insurance from year 1992 to year 2001. Keywords: Reinsurance, XL premiums, large claims, Extreme Value Modeling, Bayes, Exact Credibility, Pareto distribution, Generalized Inverse Gaussian family. 1-Introduction This paper addresses the Bayesian estimation of the shape parameter of Pareto distributions, and its application to the pricing of large claims XL reinsurance contracts. It is well known that modeling the tail of the claim severity distribution is of the utmost importance when dealing with large claims. Extreme Value Theory (EVT) gives clues for either determining a threshold defining the class of large claims or imposing by means of an asymptotic argument, a distribution function to the excesses over that threshold (see for instance Beirlant, Teugels, Vynckier (1996), Embrechts, Klüppelberg, Mikosch (1997), Reiss, Thomas (2001), and McNeil (1997)). This facilitates the tail modeling through a Generalized Pareto Distribution (GPD) making possible the subsequent reinsurance premium calculations. We will only focus on the most dangerous case when the GPD has a positive shape parameter 0 ξ > , corresponding to a Pareto distribution. In this paper we will consider XL contracts with a priority equal to the excess threshold and an unlimited cover. We will also distinguish among two kinds of reinsurance premiums. A collective premium is calculated starting from a sample